1. Field of the Invention
The invention pertains to modelling and simulation generally and in particular to the modelling of the interaction of rigid bodies.
2. Description of the Prior Art
One use of computers is to develop systems which model the physical world. Such systems provide electronic laboratories for a variety of applications. For example,
As an integral part of Software Prototyping, i.e. the building and verification of electro-mechanical product designs on the computer (CAD/CAM). PA1 Simulation of automatic assembly operations that might also involve relative sliding of the mating parts. PA1 The design of automatic parts orienting machines and strategies. PA1 As a testbed for verifying control strategies for robotic and other physical systems. For designing walking, hopping and other mobile robots; and for simulating various kinds of robotic activity. PA1 Simulation of activity in hazardous and exotic environments like in deep sea, in space, or on other planetary bodies. PA1 For studying behavior of granular assemblies and granular flow. PA1 For studying rock mechanics, avalanches and other geological activity, building stability, and the effects of explosions on structures. PA1 In Biomechanics and safety studies, especially for vehicle crash simulation. PA1 As an educational tool for building insight into dynamical systems, whose behavior can sometimes be quite counter-intuitive. For example, motion due to gyroscopic loads, effects of friction, etc. PA1 Modeling molecular interactions. PA1 Realistic animation for computer graphics and entertainment. PA1 For building a computer model of the physical world that a `user` could interact with using sensory interfaces, commonly known as virtual reality.
A common thread in all these applications is the need for realistic and efficient modelling of the contact between objects. An additional, anthropocentric, reason for modeling such contact is its importance in human tactile activity (touching, rubbing, grasping, etc.). Similarly, the correct incorporation of contact loads in feedback control algorithms can be a critical element for autonomous robots and automatic parts orienting machinery.
Contact between bodies can occur in two situations. The bodies may be constrained together in the form of a joint or mechanism. Or they may interact through collisions (impacts) and sustained gentler contacts. Classically, an impact between a pair of bodies is modeled as a phenomenon with instantaneous transfer of momentum and energy. Sustained contact occurs when bodies remain in stationary contact or slide or roll along each other.
Of the two kinds of physical interactions cited above, mechanisms are easier to model because the motion resulting due to a joint (or hinge) is translatable into holonomic constraints that can be generated automatically and incorporated into the equations of motion. On the other hand, modeling arbitrary impacts and/or sustained contact is difficult because of: (a) the lack of a concise theory for three-dimensional (3-D) impact with friction, (b) complex nonholonomic constraints associated with the motion of bodies that move while maintaining contact, the time discontinuity of these constraints, and, (c) issues related to friction.
Given inertial properties (masses and moments of inertia) and external loads (forces and moments), the problem of simulating motion of interacting bodies reduces to the correct determination of contact loads, or equivalently, efficient computation of `contact mechanics`. In this context, contact mechanics divides into two parts: the geometric part that deals with the detection of contact between objects and the dynamics part that deals with the estimation of forces at these contacts. The first step in the calculation of contact forces involving formulation of a contact model.
In full detail, the interaction between bodies during contact is a complex phenomenon that involves local deformations and acoustic waves propagation throughout the bodies, fracture, dissipation of energy through internal friction and heat, plastic deformation, sound, and other mechanisms. There does not exist any complete and elegant formulation to describe this phenomenon in its entirety. All practical models use some idealized approximation to the actual contact phenomenon. Classical contact models were designed to avoid the need for tedious calculation such as that requiring numerical integration. The price of this avoidance was suppressed detail, reliance on parameters such as the coefficient of restitution that did not always correspond well to material constants, and a proliferation into cases. But with the advent of high speed computers, computation intensive models have also become feasible.
From a computational viewpoint, an important criterion for evaluating contact models is their time complexity in estimating forces. This includes the computation required for detecting and analyzing contact and that required for calculating forces. In fact, computational complexity is the main limiting factor in determining the `size` of the problem that can be simulated in reasonable time. Consequently, the prior art has had to trade off between detail (essential for an accurate simulation) and speed. It is an object of the invention disclosed herein to provide modelling apparatus which gives a better trade-off between detail and speed than has been available heretofore.